Black Hole Engineering

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Ah, black holes. Flaws in the fabric of the universe. Empty voids from which nothing can return. The ultimate unknowable mystery.

But what are they good for?

Basics

Lets start with a brief introduction to black holes.

Things like planets and stars and other massive bodies have gravitational fields around them that tend to draw things toward them and trap stuff on them. In order to get away from such a body, you need to shoot yourself off it with a speed higher than its escape velocity. If you don't have that much speed, you can't get away. When you pack enough mass into a small enough volume, its gravity gets so high that the escape velocity is higher than the speed of light. Because nothing can go faster than light, nothing can escape. This is a black hole.

That's the description motivated by Newtonian gravity, anyway. But when gravity gets really strong Newtonian gravity breaks down and you need to use general relativity instead. Curiously, the size and mass where light (and everything else) is trapped is the same as the Newtonian case. But instead of light and other things flying out, looping around, and coming back space-time gets strange. At the critical distance where light would be trapped you get a surface called an event horizon. Nothing that passes into an event horizon can ever get back out again. The gravity at and inside the event horizon is so strong that it rotates space and time enough that the direction inwards toward the center becomes your inevitable future. You can no more resist going toward the middle of the hole that you can avoid seeing what fate awaits you.

An uncharged and non-rotating black hole at rest is described by the Schwarzschild geometry. The radius of its event horizon is the Schwarzschild radius

r_S = 2 G M / c²

where M is the mass of the black hole, G is the gravitational constant, and c is the speed of light in vacuum. As an example, a black hole with a mass of 50 million metric tons would have a Schwarzschild radius of 7.43 × 10^-17 meters. This is slightly under one-tenth the radius of a proton.

Charged and/or rotating black holes get more complicated.

Energy

Hawking radiation

Famously, noting that goes into a black hole can ever come back out again. But something comes out. For it turns out that black holes have a temperature and that, like everything with a temperature, they radiate electromagnetic radiation. In fact, being perfectly black, they radiate as a perfect black body. This radiation is called Hawking radiation after its discoverer, physicist Stephen Hawking. For normal sized black holes, those the size of stars or galaxies, this temperature is very small and the radiation power is absolutely minuscule. But the smaller the hole, the hotter it gets and the more power it radiates. For a black hole with mass M, the Hawking temperature T_H is

T_H = ℏ c³ / (8 π G k_b M)

where ℏ is Planck's constant, π is the circle constant, and k_b is Boltzmann's constant. Curiously, this means that the wavelengths around the peak emission of light in its spectrum is near the size of its event horizon. The power radiated by a hole of this temperature is

P_H = ℏ c⁶ / (15360 π (G M)³).

The radiated energy comes from the black hole's mass-energy, so a black hole will shrink over time as its mass is radiated away. As the mass decreases, the temperature goes up and so does the power output. So you get a runaway process of the hole getting hotter and hotter and radiating more and more power until POOF! It's gone in a flash of light and radiation. The lifetime remaining of any black hole, assuming more mass doesn't fall into it, is

t_H = 5120 π G² M³ / (ℏ c⁴).

This is a neat result. It allows perfect conversion of mass-energy into radiant energy. However, the actual implementation can get a bit inconvenient.

Let's skip for the moment the details of how you get a black hole. We'll assume that you have a magic black hole making box that can pop out whatever size of hole you need. Now let's say you want a megawatt of power. What size of hole do you need? It turns out to be a cool 18.8 billion metric tons. A hole that size is rather hard to carry around with you. And its temperature will be 6.5 billion kelvin. At that temperature its radiation is primarily hard x-rays and gamma rays. On the plus side, it's about 3000 times smaller in radius than a typical atom. So you could slip it into your pocket; just don't expect it to stay there.

Here we see one of the issues on trying to utilize Hawking power from black holes. Usable amounts of power generally come with horrendous power to mass ratios with the energy released as highly penetrating ionizing radiation. And if you start getting to masses that are more practical to deal with, you've got more of a bomb than a reactor – a 1000 ton black hole will release all of its 85 gigatons TNT equivalent in just a little under a minute and a half.

Let's take an example of a black hole with a mass of 50 million metric tons, for reasons that will become clear later. We have already found that this hole is only about a tenth the size of a proton. But that tiny speck of compact mass has a temperature of 2.45 × 10¹² kelvin. It puts out a radiated power of 1.425 × 10¹¹ watts, which is a rate of mass loss of 1.585 micrograms per second. Or in somewhat more descriptive terms, about the energy released by the detonation of 35 tons of TNT every second. Left to its own devices, it will slowly get brighter and brighter, losing mass faster and faster, until it eventually radiates itself away in 333 million years.

Penrose process

Penrose batteries

How much energy can you get out? If you start with a Schwarzschild hole, can you turn it in to a Kerr hole and extract more energy out of it than the original Schwarzschild hole? (Probably not, Hawking's area theorem and all).

Feeding a black hole

If you are extracting energy from a black hole, you might want to eventually put that energy back in to avoid using up your black hole too soon. You can do this by letting mass or other forms of energy fall into the hole, passing through its event horizon to get trapped forever.

Tidal disruption

If you have something smaller in size than a black hole's event horizon and you drop it straight in, it should enter the hole without any particular complications. But as the object approaches the hole, the hole's changing gravity will affect different parts of the object differently. Gravity drops off with distance, so the parts of the object nearest the hole will be getting pulled harder than those furthest away. This means that once you account for the average force on the object accelerating it toward the hole, you have an additional force acting on the body to tear it apart along the direction to the hole. Meanwhile the direction of gravity is toward the center of the hole, pointing radially inward. Again, after accounting for the average force on the object this means that the parts furthest to the left are experience a residual force pointing to the right and vice versa. So the net result is that tidal forces stretch an object along the direction towards the center of the hole and squish it together in the directions transverse to that direction. This is called "spaghettification".

Tidal forces fall off faster than the average force of gravity on an object. Whereas gravity falls off with the square of the distance, tides fall off with the cube of the distance. So far out from a black hole, you might be falling comfortably but as you get closer the tides get strong quickly. Very large black holes, like the supermassive black holes at the center of galaxies, might not generate any noticeable tides even as you fall though the event horizon. Smaller holes, on the scale of stellar mass black holes, do generate enough tides to spaghettify any astronaut unlucky enough to fall into them.

Accretion disks and astrophysical jets

If the thing you drop into a black hole isn't dropping straight in – maybe it has a bit of transverse velocity as it gets sucked down – it is likely to miss the event horizon and slingshot around on an orbit. However, even as it misses the all-devouring beast at the center tidal disruption is still pulling the object apart. A close enough approach will have the tides rip apart the object and smear it out into a smudge of debris. The inner parts of the debris cloud will be orbiting faster than the outer parts, leading to shear flow and friction and drag. This leads to heating of the debris, coming from the object's kinetic energy. After enough passes, the former object will get spread out into a ring around the hole. The closer the debris is to the hole, the faster the difference in speed between adjacent streamlines and the more heating will occur. So you can get the inner parts of the ring glowing brightly with radiated heat.

Most physical process that can feed matter into a black hole start with the infalling matter having some angular momentum. Because the angular momentum is conserved it naturally results in accretion disks forming as the matter falls in.

As the inner part of the disk radiates heat, it loses kinetic energy and gets a little bit closer to the event horizon. As it gets closer it gains heat at a greater rate and its temperature increases. When it gets hot enough, the matter turns into a plasma. To a good approximation, plasmas cannot cross magnetic field lines. A strong field with a diffuse plasma will have the plasma move along the field line direction. A dense, fast plasma, on the other hand, can bully through weak field lines, stretching out the field so that it moves with the plasma. In a turbulent plasma, or, in tis case, a circulating plasma, the field gets stretched out enough that it can come back and meet itself, getting stronger and stronger. This dynamo effect will amplify even very weak fields within the accretion disk, forming a strong magnetic field near the black hole.

And this is where things get a bit weird. Something happens – we're still not entirely sure what – and the interaction of the strong field with the energetic plasma right near the event horizon creates jets of fast moving plasma, high energy particles, and electromagnetic radiation shooting out along the axis of the accretion disk, usually in both directions.

The accretion process can extract somewhere between 10% and 50% of the mass energy of infalling matter into radiated energy and energy of the jets. If this energy can be collected, it can provide an additional source of energy beyond what you can get from Hawking radiation and its somewhat inconvenient limits. So now we must see what limits the rate of accretion to see how much energy we can get out of it and also how fast we can recharge our hole for the extraction of Hawking and Penrose energy.

Mass collection rates

Suppose you have a black hole inside of some material. This might be a rock, or a star-hot plasma, or the diffuse gas of interstellar space.

If you are at rest with respect to the surrounding material, you'll get that material falling toward you. It will pile up as it crams together trying to get to the hole, until you reach a point where the flow turns super-sonic and the material free-falls the rest of the way into the hole. Finding the feeding rate is thus a choked flow problem.

If the hole is moving through the material faster than the speed of sound, material passing close to the hole will get deflected by the hole's gravity to converge in a wake behind it. Where it collides with other gas coming in from all directions in the wake, the gas comes to a halt and from there it can freely fall into the hole from behind.

The analysis of these two limits may be combined to give the Bondi-Hoyle accrection rate

ṁ_BH = 4 π ρ G² M²/ (c_s² + v²)^3/2

where ρ is the density of the stuff the hole is in, c_s is the speed of sound in the medium, and v is the speed of the hole through the medium. The distance at which the in-falling material goes from subsonic choked flow to supersonic free-fall is the Bondi radius

r_B = 2 G M / c_s³.

If the Bondi-Hoyle accretion rate is too low, the black hole will be losing matter faster to Hawking radiation than it will be gaining mass to accretion. This depends on the variables described above, but let's look at what happens if we drop it into solid rock. Assuming a typical density of rock of 2.7 grams per square centimeter and a sound speed in rock of about 5 kilometers per second, we find that holes that are larger than 43 million metric tons are able to absorb a net gain in mass while those below this limit lose more mass to Hawking radiation than they gain by eating the rock. If you want to feed your hole with rock, you'll need it to be bigger than 43 million metric tons.

The best material for feeding your black hole, according to the Bondi-Hoyle accretion rate, is the heavy metal thallium. If you drop your hole into a blob of thallium, it can achieve a net mass gain at a mass of only 35 million metric tons. For black hole masses below this, you cannot feed a black hole on normal matter at room temperature and pressure (whether it can feed at the crazy high pressures at the cores of planets or stars is a subject not explored here).

Radiation pressure

Both the Hawking radiation and the radiation from the accretion disk will be shining out of an accreting black hole. This radiation will encounter material from the accretion disk. The radiated light can scatter off electrons in the disk material; on average, this will push them outward. The electrons will then drag any assorted atomic nuclei in the disk material with them. This puts a limit on how much material can flow into the black hole – if it is too bright, it will push everything away. If the hole gets brighter than this limit, it can no longer feed.

This is often referenced in terms of the Eddington luminosity

L_E = 4 π G M (A/Z) m_p c / σ_T

where A is the average atomic weight of the plasma, Z is the average atomic number, m_p = 1.672622 × 10^-27 kg is the mass of a proton, and σ_T = 6.65246 × 10^-29 m² is the Thompson cross section for scattering light off an electron. If something is shining with the Eddington luminosity, it will keep matter from falling in. Strictly speaking, this assumes hydrostatic equilibrium; for problems that are time varying or with steady-state flows the Eddington limit does not necessarily apply. However, it is often a good first guess to figure out when the radiation chokes off the inflow in accretion disks. There are some configurations of accretion disks that can support luminosity higher than the Eddington limit, but most are at or below this limit.

If we assume that our black hole's accretion disk is Eddington limited, we can find out how big it needs to be in order to accrete any matter at all, or to achieve net mass gain after its Hawking radiation losses are accounted for. In hydrogen gas, with A/Z = 1, we find that a hole must have a mass of at least about 39 million metric tons for any matter to fall in past the Hawking radiation pressure. The hole's mass has to be in the 40 to 44 million metric ton range to gain mass via accretion faster than it is lost to Hawking radiation, depending on the efficiency at which matter in the accretion disk is converted into radiation. If you drop the hole into rock or other light elements you'll have an A/Z ratio of 2 or very slightly higher. Setting A/Z = 2, we find that you can't get any accretion for masses under 31 million metric tons and, again depending on the radiative efficiency of the accretion disk, you need somewhere in the range of 32 to 35 million metric tons to reach breakeven in terms of mass loss versus mass gain. Even for very heavy elements like lead or uranium, with an A/Z ratio of approximately 2.5, you need at least 29 million metric tons to accrete matter at all and somewhere between 30 and 33 million metric tons to break even.

In other words, if you want to be able to add mass to your black hole by having it gobble up surrounding matter, you'll want it bigger than several tens of millions of metric tons.

Interestingly, the limit for net mass gain for the Eddington limit is very similar to that of the Bondi_Hoyle limit. In order to get a black hole that gains mass, you're pretty much going to need at least a mass somewhere in the 30 to 50 million metric ton range.

Reaction rates at sub-atomic sizes

We now know the rate at which matter can fall on to a black hole, getting past both the radiation coming from the hole and its inner accretion disk and for getting past the choked flow of the material getting in its own way. But what about when it reaches the hole? Obviously, if the hole is bigger than the size of an atom any atoms it touches will immediately get sucked in. But a lot of holes of engineering interest are much smaller than this. A black hole with a mass of 50 million tons would have a Schwarzschild radius of about 10 times smaller than that of a proton. If a hydrogen atom fell into the hole, it would end up sitting there with the black hole inside of the proton. How quickly could the hole slurp up that proton and its companion electron?

It is easy enough to get an estimate of how fast a proton or neutron will get eaten once a black hole is inside of it. Both protons and neutrons have a radius of about 8.4 × 10^-16 meters. Both are made up of three quarks. This gives a quark density of about 1.21 × 10⁴⁵ per cubic meter inside of the proton or neutron. Because the binding energy of the quarks is much larger than the mass-energies of the quarks, we can assume that they are highly relativistic and are moving at about light speed. Multiply the density by the speed to get the flux (particles passing through per area per time) and multiply by the surface area of the hole to get the absorption rate of the quarks. Once one quark is eaten, color confinement ensures that the rest of the quarks cannot leave and the particle is stuck to the black hole until the rest of it is eaten, which time we can guestimate by the time needed to eat three quarks. For our 50 million ton black hole, this shakes out to about 1 × 10^-22 seconds to eat a proton or neutron. If we multiply by the mass of a proton or neutron, we find that the 50 megaton black hole can eat protons and neutrons at a rate of about 1.4 × 10^-5 kg/s if it has a constant supply of protons and neutrons ready to immediately fall into the hole once the previous one was eaten.